Haobo HuaDepartment of Mathematics,

Korea University,

Seoul 136-713, China;

Department of Mathematics and Physics,

Zhengzhou Institute of Aeronautical

Industry Management,

Zhengzhou 450015, China

e-mail: huahbo@korea.ac.kr

Jaemin Shine-mail: zmshin@korea.ac.kr

Junseok Kim1

e-mail: cfdkim@korea.ac.kr

Department of Mathematics,

Korea University,

Seoul 136-713, China

Level Set, Phase-Field, andImmersed Boundary Methodsfor Two-Phase Fluid FlowsIn this paper, we review and compare the level set, phase-field, and immersed boundarymethods for incompressible two-phase flows. The models are based on modified Navier–Stokes and interface evolution equations. We present the basic concepts behind theseapproaches and discuss the advantages and disadvantages of each method. We also pres-ent numerical solutions of the three methods and perform characteristic numericalexperiments for two-phase fluid flows. [DOI: 10.1115/1.4025658]

Keywords: level set method, phase-field method, immersed boundary method, two-phaseflows, surface tension, Navier–Stokes equation, computational fluid dynamics

1 Introduction

Many important industrial problems involve multiphase fluidflows [1–4]. The multiphase fluid flows are related to many physi-cal problems, such as the non-Newtonian flow, microtube flow,droplet impact on a solid surface, and Marangoni convection.However, the modeling and simulation of multiphase flows aredifficult because of the moving interfaces.

We consider two incompressible and immiscible fluids in atwo-dimensional domain X ¼ X1 [ X2 and define the interface asC ¼ X1 \ X2 (see Fig. 1). The equations governing the motion ofthe two fluids are the modified Navier–Stokes equations:

q@u

@tþ u � ru

� �¼ �rpþr � g ruþruT

� �� �þ SFþ qg (1)

r � u ¼ 0 (2)

where uðx; tÞ is the velocity, pðx; tÞ is the pressure, qðx; tÞ is thedensity, gðx; tÞ is the viscosity, g is the gravitational force, thesuperscript T denotes the transpose, x ¼ ðx; yÞ are Cartesian coor-dinates, and t is the time variable. The continuum surface force(CSF) model [5] is used to represent the singular surface tensionforce:

SF ¼ �rjdCn (3)

where r is the surface tension coefficient, j is the mean curvatureof the interface, dC is the surface d function, and n is the unitnormal vector to the interface from fluid 1 to fluid 2.

We focus on the review of three methods for simulating two-phase flows. The level set method (LSM) [6–35] uses a level setfunction to capture the interface. The phase-field method (PFM)[36–67] uses an order parameter to capture the interface, and theimmersed boundary method (IBM) [68–91] uses Lagrangianmarker points to track the interface. We will provide a detaileddescription of the basic techniques of each method.

LSM, first introduced in Ref. [6], is a popular computationaltechnique for capturing moving interfaces (see Refs. [16,20] for

reviews). For the two-phase flows, the authors in Ref. [8] firstintroduced a reinitialization procedure to maintain the level setfunction as a distance function. In this method, a smoothed d func-tion is used to adapt the CSF framework.

PFM is an increasingly popular method for modeling thedynamics of multiphase fluids (see Refs. [39,66] for reviews). InPFM, there is a diffuse-interface with a finite width between twophases. The density, viscosity, and other physical quantities arecharacterized by an order parameter, which is governed by themodified Cahn–Hilliard equation. The Cahn–Hilliard equationwas first proposed by Cahn and Hilliard to describe the initialstage of spinodal decomposition [92] and it models the interfacedynamics, including the surface minimization, topologicalchanges, and phase separation.

IBM was originally developed to model the blood flow in theheart [68,69]. Since then, a number of modifications and improve-ments have been proposed (refer to Refs. [78,81] for reviews).The authors in Ref. [70] used the d function to construct an indica-tor function to distinguish the fluid properties. The interface isalso numerically diffused by the use of d function to efficientlyand conveniently distribute boundary forces.

The paper is organized as follows: 1) In Sec. 2, we present thegoverning equations for LSM, PFM, and IBM, 2) We summarizethe formulas of surface tension force in Sec. 3, 3) We describe thetreatment for the variable density and viscosity functions in Sec.4, 4) We discuss volume conservation problems in Sec. 5, 5) InSec. 6, we present the numerical method to solve the discreteNavier–Stokes equations and respective equations for the inter-face, 6) Numerical results are presented in Sec. 7, and 7) Finally,conclusions are drawn in Sec. 8.

Fig. 1 Schematic of a two-phase domain

1Corresponding author.Contributed by the Fluids Engineering Division of ASME for publication in the

JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 7, 2013; final manuscriptreceived October 8, 2013; published online November 22, 2013. Assoc. Editor:Samuel Paolucci.

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2 Governing Equations and Interface Representation

The three methods share common modified Navier–Stokesequations (1) and (2). In this section, we briefly describe the levelset function, phase-field function, and Lagrangian variable forrepresenting the interface of the two immiscible fluids. We thenintroduce the corresponding governing equations for evolving theinterface. We also briefly discuss the advantages and disadvan-tages of each method.

2.1 LSM. The interface of two phases is defined implicitlyusing the level set function /ðx; tÞ. Here, / is taken to be thesigned distance from the interface C and it becomes the distancefunction satisfying jr/j ¼ 1. We take the value of / to be zero atthe interface C. Thus, / has an opposite sign in each phase; seeFig. 2.

The evolution of the level set function / is governed by thetransport equation

/t þ u � r/ ¼ 0 (4)

During the process of interface evolution, / tends to deviate fromthe signed distance function. However, because the density andsurface tension depend on /, we should maintain / as the signeddistance function [8,10]. Therefore, at each time step, a reinitiali-zation step is needed to recover to the signed distance functionwithout changing its zero level set. Specifically, taking the func-tion / at time t as an initial condition, we solve the reinitializationequation to the steady state

@d

@sðx; sÞ ¼ Sð/ðx; tÞÞ 1� rdðx; sÞj jð Þ (5)

dðx; 0Þ ¼ /ðx; tÞ (6)

where s is the pseudotime and Sð/Þ is the sign function. For thenumerical implementation, we use a smoothed sign function

Sbð/Þ ¼ /=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2 þ b2

p, where b is usually one or two grid lengths.

For more details, we refer to Ref. [19]. After solving up to thesteady state, /ðx; tÞ is replaced by dðx; ssÞ, where ss is the steady-state time. There are efficient ways to solve Eqs. (5) and (6) to thesteady state by using the fast marching method [11].

Advantages of LSM include the simplicity to implement, abilityto capture the merging and breakup of interfaces automatically,and flexibility to describe the complex interface geometry. How-ever, typical LSM has a lack of mass conservation. This couldlead to nonphysical interface motions, which severely deterioratethe accuracy and stability of the simulation results. Therefore,many approaches have been developed to resolve the mass conser-vation problem for LSM, which will be described in Sec. 5.1.

2.2 PFM. In PFM, a thin finite interface is used to separatetwo homogeneous phases. This method uses an order parameter /that is a measure of the relative composition or volume fraction ofthe two components. The function / is distributed continuously

on thin interfacial layers and uniformly in the bulk phases asshown in Fig. 3. Here, the order parameter is defined by / � 1 influid 1 and / � �1 in fluid 2, while the interface C is defined by/ ¼ 0. The sharp fluid interfaces are replaced by thin (but non-zero) thickness transition regions.

The evolution of the phase-field function / is governed by theadvective Cahn–Hilliard equation as follows:

@/@tþr � ð/uÞ ¼ MDl (7)

l ¼ F0ð/Þ2 � e2D/ (8)

where M is the constant mobility, l is the chemical potential,Fð/Þ ¼ ð1� /2Þ2=4 is the bulk energy density that has two min-ima corresponding to the two stable phases of the fluid, and e is ameasure of interface thickness. It is convenient to use the dimen-sionless Peclet number, which is defined as Pe ¼ UcLc=M, whereUc is the characteristic velocity and Lc is the characteristic length(refer to Ref. [52] for details).

As shown in Fig. 4, from the choice of an equilibrium profile

/ðxÞ ¼ tanhðx=ðffiffiffi2p

eÞÞ on the infinite domain, the concentrationfield varies from �0:9 to 0:9 over a distance of about

n ¼ 2ffiffiffi2p

e tanh�1ð0:9Þ [41]. We note that n ¼ OðeÞ, and, in thesharp interface limit e! 0, the classical Navier–Stokes equationsand jump conditions are recovered [39].

PFM shares the advantages of LSM. Moreover, because theorder parameter has physical meanings, it can be applied tomany physical phase states such as miscible, immiscible, and par-tially miscible ones. However, the phase-field functions changequickly near the interface and, hence, must be well resolved. Inother words, a relatively large number of grid points near theinterface are needed. Choosing appropriate parameter e value isimportant for the accurate calculations. Taking too large e can

Fig. 2 (a) Zero contour of the signed distance function / and(b) surface plot of / with the zero contour

Fig. 3 (a) Zero contour of the order parameter / and (b) sur-face plot of / with the zero contour

Fig. 4 Concentration profile across the diffused interfaceregion with the thickness n

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make nonphysical solutions, while too small e leads to numericaldifficulties.

2.3 IBM. In IBM, the interface C between two fluids isdescribed by the parametric function Xðs; tÞ, where 0 � s � L andL is the length of the boundary (see Fig. 5). The fluid velocityuðx; tÞ is an Eulerian variable, whereas the interface boundaryvelocity Uðs; tÞ is defined at the Lagrangian variable Xðs; tÞ, byusing the d function and velocity uðx; tÞ. The two-dimensionalDirac d function d2ðxÞ is defined by the product of one-dimensional Dirac d functions, d2ðxÞ ¼ dðxÞdðyÞ.

The evolution of the interface is governed by

@Xðs; tÞ@t

¼ Uðs; tÞ (9)

Uðs; tÞ ¼ð

Xuðx; tÞd2ðx� Xðs; tÞÞdx (10)

First, the flow is computed in the computational domain withsurface tension forces generated by the immersed boundary.The fluid reacts to the existence of the boundary via the singu-lar force, which is spread to the surrounding Eulerian pointsusing a discrete d function. The immersed boundary is thenadvected by the fluid velocity obtained from interpolating theEulerian points.

IBM has the principal advantage over the other two methodsbecause it can use a large number of interfacial marker points toaccurately handle the interface geometry. The major drawback isthe difficulty of topological change unless the additional work forgeometric problems is considered. Because the interface betweentwo fluids moves discretely, area conservation does not, in gen-eral, hold. We will discuss the mass conservation problem forIBM in Sec. 5.3.

2.4 Multiphase and Multicomponent System. We brieflydescribe the multiphase and multicomponent system for eachmethod. Let us consider N-phase fluid flows. In LSM, the motionof the interfaces is given by

@/i

@tþ u � r/i ¼ 0 for i ¼ 1;…;N (11)

where the separate level set function /i, which is positive in phasei and negative outside; please refer to Refs. [15,18,23] for moredetails. In PFM, let c ¼ ðc1; c2;…; cN�1Þ be the phase variablewith c1 þ c2 þ � � � þ cN ¼ 1 [66,67]. The motion of the interfacesis given by

@c

@tþr � ðcuÞ ¼ MDl (12)

l ¼ F0ðcÞ � e2D/ (13)

In IBM, the evolution of the interface is governed by

dXi

dt¼ Ui (14)

where Xi represents the position of a marker point on the interfaceCi and Ui is the velocity at Xi. Please refer to Refs. [75,77] formore details.

3 Surface Tension Force

We describe how to define the surface tension force for LSM,PFM, and IBM.

3.1 LSM. In LSM, the surface tension force is given by

SF ¼ �rr� r/jr/j

� �dað/Þ

r/jr/j (15)

Here, / is the level set function, and the interface curvature jð/Þcan be calculated by jð/Þ ¼ r � r/=jr/jð Þ. r is the surfacetension coefficient. Using Ref. [68], the smoothed d functiondað/Þ [8,9] is defined as

dað/Þ ¼1

2aþ 1

2acos

p/a

� �; if j/j � a;

0; otherwise

8<: (16)

3.2 PFM. In PFM, a surface tension force formulation, basedon CSF, is given by

SF1 ¼ �3ffiffiffi2p

4rer � r/

jr/j

� �jr/jr/ (17)

Here, / is the phase-field function, and e is a small positiveparameter in Eqs. (7) and (8). SF1 comes from the CSF frame-work [54]. Similar to the surface tension formula in LSM,r � r/=jr/jð Þ accounts for the interface curvature jð/Þ. Wenote that some other surface tension force formulas driven bythermodynamics are frequently used in PFM for two-phase flows.These include:

SF2 ¼3ffiffiffi2p

4rer � jr/j2I �r/�r/

(18)

SF3 ¼3ffiffiffi2p

r4e

lr/ (19)

SF4 ¼ �3ffiffiffi2p

r4e

/rl (20)

where I is the unit tensor dij and the term r/�r/ is the usualtensor product defined by ðr/�r/Þij ¼ ð@/=@xiÞð@/=@xjÞ

In SF2, r/�r/ is induced elastic stress due to the mixing ofthe different species [36,48]. SF3 was derived in Ref. [38]. SF4,chemical potential formulation, was proposed by Ref. [41].The formulas for SF that have been applied in the literatureare as follows: SF1 [54,60,61], SF2 [36,40,44,45,48,52,], SF3

[37,38,43,46], and SF4 [41,42,49,58,60].

3.3 IBM. In IBM, the surface tension force is given by

SFðx; tÞ ¼ð

Cfðs; tÞd2ðx� Xðs; tÞÞds (21)

Fig. 5 Immersed boundary configuration Xðs; tÞ for represent-ing the interface C

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fðs; tÞ ¼ r@2Xðs; tÞ@s2

(22)

where fðs; tÞ is the boundary force density. Note that@2Xðs; tÞ=@s2 accounts for the interface curvature j.

4 Variable Density and Viscosity

The density and viscosity are constant within a fluid but cantake different values in each phase. Here, the three methods use asimilar tactic to identify the different phases. Let qi and gi, i ¼ 1,2, be the density and viscosity of fluid i, respectively. The densityand viscosity values are determined by

q ¼ q2 þ ðq1 � q2ÞH (23)

g ¼ g2 þ ðg1 � g2ÞH (24)

where H is the Heaviside (step) function whose value is one influid 1 and zero in fluid 2. We illustrate the three discrete Heavi-side functions used in the three methods in Fig. 6.

4.1 LSM. For LSM, because the jump in phase propertiesacross the interface may lead to numerical difficulties, it needs tobe smoothed [8]. Hence, Hð/Þ is replaced by a smoothed Heavi-side function as

Hað/Þ ¼

0; if / < �a;1

21þ /

aþ 1

psin

p/a

� �� �; if j/j � a;

1; if / > a

8>><>>: (25)

where 2a corresponds to the interface thickness (a > 0). Note thatH0að/Þ ¼ dað/Þ. The regularized density and viscosity are thendefined as

qð/Þ ¼ q2 þ ðq1 � q2ÞHað/Þ (26)

gð/Þ ¼ g2 þ ðg1 � g2ÞHað/Þ (27)

The interface thickness depends on the grid size h and should bechosen as small as possible for accuracy but large enough to stabi-lize the system. When a is relatively small, e.g., a ¼ 0:5h, theinterface tends to be staggered. In contrast, if a is relatively large,e.g., a ¼ 3:0h, the interface tends to be smeared. The values of athat have been applied in the literature are as follow: 0:5h [24], h[33], 1:5h [8,21,26], 2h [12,28], 2:5h [9], and 3h [12].

4.2 PFM. The density and viscosity are linear functions ofthe phase field [47,54]. The density and viscosity of the mixtureare defined as

qð/Þ ¼ q2 þ ðq1 � q2Þ1þ /

2(28)

gð/Þ ¼ g2 þ ðg1 � g2Þ1þ /

2(29)

The harmonic interpolation for the variable density and viscositycan also be used [48] as

1

qð/Þ ¼1

2

1þ /q1

þ 1� /q2

� �(30)

1

gð/Þ ¼1

2

1þ /g1

þ 1� /g2

� �(31)

4.3 IBM. For IBM, we introduce an indicator function Hðx; tÞ[70], which has the characteristics of the Heaviside function. Letus define a gradient field

rHðx; tÞ ¼ �ð

CnðXðs; tÞÞd2ðx� Xðs; tÞÞds (32)

which is nonzero near the interface and zero in the other domain.To find the indicator function, we solve Poisson’s equation

DHðx; tÞ ¼ �r �ð

CnðXðs; tÞÞd2ðx� Xðs; tÞÞds (33)

with the Dirichlet boundary condition. Then, the variable fluidproperties q and l can be represented by

qðHÞ ¼ q2 þ ðq1 � q2ÞH (34)

gðHÞ ¼ g2 þ ðg1 � g2ÞH (35)

5 Volume Conservation

5.1 LSM. One of the main drawbacks of LSM is its lack ofmass conservation. This leads to nonphysical motions of the inter-face, which severely deteriorate the accuracy and stability of thesimulation results [10]. Several methods for improving mass con-servation have been proposed. The authors in Refs. [13,22,32]take full advantage of LSM for better curvature approximationusing the volume of fluid (VOF) method. In Refs. [14,30], aboundary condition capturing method developed from the ghostfluid method (GFM) models the pressure jump directly, leading tono need to add the surface tension force SF. The surface tension ismodeled directly by imposing a pressure jump across the inter-face. The exact jumps in density and viscosity are allowed so thatthe nonphysical finite width interface smeared by d function canbe replaced by a sharp interface. In Refs. [17,27], the authors pre-sented a hybrid particle LSM to improve the mass conservationand interface resolution by using Lagrangian marker particles torebuild the level set in underresolved regions.

Besides the above studies, in Refs. [24,31], the authors demon-strated another way to conserve mass. Instead of using a reinitiali-zation process, they use the regularized characteristic function.An initialization step, formulated as a conservation law, is used topreserve the smooth profile. Another technique is the so-calledvolume-fraction level set approach [25], where an alternative for-mulation of the Heaviside function is used for property evaluationand mass conservation correction.

Fig. 6 (a) Smoothed Heaviside function Hað/Þ for LSM, (b) ð1þ /Þ=2 for PFM, and(c) Indicator function H for IBM.

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Recently, a geometry-based reinitialization scheme for the levelset function was compared to the widely used partial differentialequation-based scheme [34]. The authors show that the algorithmis well suited to curvilinear grids, and mass is also conserved bymeans of a localized mass correction.

5.2 PFM. As the evolution of the phase-field function / isgoverned by the fourth-order Cahn–Hilliard equation, which natu-rally conserves mass, and because there is a critical balancebetween nonlinear and diffusive terms, the solution’s thin transi-tion layers do not deteriorate dynamically [57,64]. However, inRef. [57], the authors pointed out that, even though the phase-fieldvariable / is conserved globally, simulating two-phase flows withCahn–Hilliard diffusion causes a droplet to shrink spontaneouslywhen / shifts from its expected values in the bulk phase. In addi-tion, the global mass conservation does not imply that the massenclosed by the zero contour of / remains constant.

We note that there is another approach in PFM using theAllen–Cahn equation [56,62,65]. Its implementation is easier thanusing the Cahn–Hilliard equation because only second-orderderivative term is shown in the Allen–Cahn equation.

5.3 IBM. When an exact projection method is used to solvethe Navier–Stokes equations, the velocity field on the Euleriangrid is discretely divergence-free. However, this does not guaran-tee that the velocity field interpolated by the d function is continu-ously divergence-free. This results in a loss of volume for a closedinterface [71,85]. Furthermore, because the interface moves in adiscrete manner, there is also a lack of volume conservation inthis process. In Refs. [90,91], the authors developed a volume-preserving scheme with negligibly additional computational cost.This scheme corrects the interface location normal to the interfaceso that the volume of one fluid remains constant.

When high Reynolds numbers are used, the local accuracy ofthe interface is important, and the smooth force distribution is lessdesirable. Thus, many researchers have considered that the sharpinterface is represented by using the immersed boundary, includ-ing the immersed interface methods (IIM) [93–96]. The forceterm is heavily connected with the employed discretizationschemes. Besides solving two-phase flows, there are many studiesfor simulating flows in the complex boundaries; refer to Refs.[97–101].

6 Numerical Solution

In this section, we present a series of numerical solutions on astaggered marker-and-cell mesh [102] in which pressure and indi-cator functions are stored at the cell centers and velocities aredefined at the cell edges (Fig. 7(a)). Let a computational domainX ¼ ða; bÞ � ðc; dÞ be partitioned in Cartesian geometry. Let Nx

and Ny be the number of cells in the x- and y-directions, respec-tively. We assume a uniform mesh with mesh spacingh ¼ ðb� aÞ=Nx ¼ ðd � cÞ=Ny. The center of each cell Xij islocated at xij ¼ ðxi; yjÞ ¼ ðaþ ði� 0:5Þh; cþ ðj� 0:5ÞhÞ for i¼ 1;…;Nx and j ¼ 1;…;Ny. We denote the discrete computa-tional domain Xh ¼ fxijg. In IBM, N Lagrangian pointsXn

l ¼ ðxnl ; y

nl Þ for l ¼ 1;…;N are used to discretize the immersed

boundary (Fig. 7(b)). In LSM and PFM, a grid function /ij is used.At the n-th time step, we have a velocity field un, which is

divergence-free, and a surface tension calculated by /n (or theboundary configuration Xn). We seek unþ1, pnþ1, and /nþ1 (orXnþ1) that are the solutions of the following semi-implicitscheme:

qn unþ1 � un

Dt¼ �qnun � rdun �rdpnþ1

þrd � gn rdun þ ðrdunÞT h i

þ SFn þ qng

(36)

rd � unþ1 ¼ 0 (37)

where g ¼ ð0;�gÞ, with g being the gravitational acceleration.Here, rd� and rd are the discrete divergence and gradient opera-tors, respectively. The projection method, introduced by Chorin[103], is applied to solve the incompressible Navier–Stokes equa-tions. We refer the readers to Refs. [8,12,90,91,102] for the discre-tization approach. An outline of the main procedures in one timestep is as follows:

Step 1. Compute the surface tension force SFn using /n (orXn). A detailed description of the discretization of SFn foreach method is given in Sec. 6.1. We calculate the density qn andviscosity gn using /n (or Xn). The numerical formulas for the vari-able density and viscosity are drawn in Sec. 6.2.

Step 2. Solve the Navier–Stokes equations to get unþ1 and pnþ1

from un and SFn. We first solve an intermediate velocity field ~un:

~un � un

Dtþ un � rdun ¼ 1

qnrd � gn rdun þ ðrdunÞT

h iþ 1

qnSFn þ g (38)

The resulting finite difference equations are written out explicitly:

~uniþ1

2;j ¼ un

iþ12;j � Dtðuux þ vuyÞniþ1

2;j þ

Dt

qniþ1

2;j

SFx�edge

iþ12;j

þ Dt

qniþ1

2;j

2ðguxÞx þ ðguyÞy þ ðgvxÞy n

iþ12;j

(39)

~vni;jþ1

2¼ vn

i;jþ12� Dtðuvx þ vvyÞni;jþ1

2þ Dt

qni;jþ1

2

SFy�edge

i;jþ12

þ Dt

qni;jþ1

2

ðgvxÞx þ ðguyÞx þ 2ðgvyÞy n

i;jþ12

�gDt (40)

where we define qniþ1

2;j¼�qn

iþ1;j þ qnij

�=2. qn

i;jþ12

, SFx�edge

iþ12;j

, and

SFy�edge

i;jþ12

are similarly defined. The advection terms are defined by

ðuux þ vuyÞniþ12;j

¼ uniþ1

2;j�un

xiþ1

2;jþ 1

4vn

i;j�12

þ vniþ1;j�1

2

þ vni;jþ1

2

þ vniþ1;jþ1

2

�un

yiþ1

2;j

(41)

and

ðuvx þ vvyÞni;jþ12

¼ vni;jþ1

2

�vny

i;jþ12

þ 1

4un

i�12;jþ un

i�12;jþ1þ un

iþ12;jþ un

iþ12;jþ1

�vn

xi;jþ1

2

(42)

Fig. 7 (a) Velocities and pressure near the cell Xij and (b)Lagrangian points Xl in the computational domain X.

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where the values �unx

iþ12;j

and �uny

iþ12;j

are computed using the upwindprocedure

�unx

iþ12;j¼

uniþ1

2;j� un

i�12;j

h; if un

iþ12;j> 0;

uniþ3

2;j� un

iþ12;j

h; otherwise

8>>><>>>:

(43)

and

�uny

iþ12;j¼

uniþ1

2;j� un

iþ12;j�1

h; if vn

i;j�12

þ vniþ1;j�1

2

þ vni;jþ1

2

þ vniþ1;jþ1

2

> 0;

uniþ1

2;jþ1� un

iþ12;j

h; otherwise

8>>><>>>:

(44)

The quantities �vnx

i;jþ12

and �vny

i;jþ12

are similarly computed. The viscos-

ity terms are defined by

2ðguxÞx þ ðguyÞy þ ðgvxÞy n

iþ12;j¼ 2

hgn

iþ1;j

uniþ3

2;j� un

iþ12;j

h� gn

ij

uniþ1

2;j� un

i�12;j

h

!þ 1

4gn

iþ1;jþ1 þ gni;jþ1 þ gn

ij þ gniþ1;j

uniþ1

2;jþ1� un

iþ12;j

h2

� 1

4gn

iþ1;j þ gnij þ gn

i;j�1 þ gniþ1;j�1

uniþ1

2;j� un

iþ12;j�1

h2þ 1

4gn

iþ1;jþ1 þ gni;jþ1 þ gn

ij þ gniþ1;j

�vn

iþ1;jþ12

� vni;jþ1

2

h2� 1

4gn

iþ1;j þ gnij þ gn

i;j�1 þ gniþ1;j�1

vni;j�1

2

� vni�1;j�1

2

h2(45)

and

ðgvxÞx þ ðguyÞx þ 2ðgvyÞy n

i;jþ12

¼ 1

4gn

iþ1;jþ1 þ gni;jþ1 þ gn

ij þ gniþ1;j

vniþ1;jþ1

2

� vni;jþ1

2

h2� 1

4gn

iþ1;j þ gnij þ gn

i;j�1 þ gniþ1;j�1

�vn

i;jþ12

� vni�1;jþ1

2

h2þ 1

4gn

iþ1;jþ1 þ gni;jþ1 þ gn

ij þ gniþ1;j

uniþ1

2;jþ1� un

iþ12;j

h2

� 1

4gn

iþ1;j þ gnij þ gn

i;j�1 þ gniþ1;j�1

uniþ1

2;j� un

iþ12;j�1

h2

þ 2

hgn

i;jþ1

vni;jþ3

2

� vni;jþ1

2

h� gn

ij

vni;jþ1

2

� vni;j�1

2

h

!(46)

We then solve the following equations for the advanced pressurefield at the (nþ 1)-th time step:

unþ1 � ~un

Dt¼ � 1

qnrdpnþ1 (47)

rd � unþ1 ¼ 0 (48)

Applying the discrete divergence operator and divergence-freeEq. (48) to Eq. (47), we obtain Poisson’s equation for the pressureat the time (nþ 1):

rd �1

qnrdpnþ1

� �¼ 1

Dtrd � ~un (49)

where

rd �1

qnrdpnþ1

� �ij

¼pnþ1

iþ1;j � pnþ1ij

h2qniþ1

2;j

�pnþ1

ij � pnþ1i�1;j

h2qni�1

2;j

þpnþ1

i;jþ1 � pnþ1ij

h2qni;jþ1

2

�pnþ1

ij � pnþ1i;j�1

h2qni;j�1

2

(50)

and

rd � ~unij ¼

~uniþ1

2;j� ~un

i�12;j

hþ

~vni;jþ1

2

� ~vni;j�1

2

h(51)

The boundary condition for the pressure is

n � rdpnþ1 ¼ n ��� qn unþ1 � un

Dt� qnðu � rduÞn

þ�gn rdun þ ðrdunÞT �

þ SFn þ qng

�(52)

where n is the unit normal vector to the domain boundary. Forexample, if we use a periodic boundary condition on the horizon-tal boundaries and Dirichlet condition on the top and bottomboundaries, then Eq. (52) is reduced as

n � rdpnþ1 ¼ n � SFn þ qngð Þ (53)

The resulting linear system of Eq. (49) is solved using a multigridmethod [104], specifically V-cycles with a Gauss–Seidel relaxa-tion. The divergence-free velocities unþ1 and vnþ1 are thenobtained using Eq. (47):

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unþ1iþ1

2;j¼ ~un

iþ12;j �

Dt

qniþ1

2;jh

�pnþ1

iþ1;j � pnþ1ij

�(54)

vnþ1i;jþ1

2

¼ ~vni;jþ1

2� Dt

qni;jþ1

2

h

�pnþ1

i;jþ1 � pnþ1ij

�(55)

Step 3. Once the updated fluid velocity unþ1 has been deter-mined, we can find the new level set and phase-field /nþ1 (orXnþ1). We will describe this in Sec. 6.3.

This completes a single time step.

6.1 Discretization of Surface Tension Force. In this section,we present the discretization of the surface tension forces.

6.1.1 LSM. In LSM, the surface tension force is given by

SFnij ¼ �rrd �

m

jmj

� �ij

dað/ijÞrd/ij

rd/ij

(56)

Vertex-centered normal vectors are obtained by differentiating thelevel set in the four surrounding cells. For example, the normalvector at the top right vertex of cell Xij is given by

miþ12;jþ1

2

¼ mxiþ1

2;jþ12;my

iþ12;jþ1

2

¼�

/iþ1;jþ/iþ1;jþ1�/ij�/i;jþ1

2h;/i;jþ1þ/iþ1;jþ1�/ij�/iþ1;j

2h

�(57)

The curvature is calculated at cell centers from the vertex-centered normals [5] and is given by

jð/ijÞ ¼ rd �m

jmj

� �ij

¼ 1

2h

mx

iþ12;jþ1

2

þ my

iþ12;jþ1

2

jmiþ12;jþ1

2j þ

mxiþ1

2;j�1

2

� my

iþ12;j�1

2

jmiþ12;j�1

2j

�mx

i�12;jþ1

2

� my

i�12;jþ1

2

jmi�12;jþ1

2j �

mxi�1

2;j�1

2

þ my

i�12;j�1

2

jmi�12;j�1

2j

!(58)

The cell-centered normal is the average of the vertex normals:

rd/ij ¼miþ1

2;jþ1

2þmiþ1

2;j�1

2þmi�1

2;jþ1

2þmi�1

2;j�1

2

4(59)

6.1.2 PFM. In PFM, the surface tension force is given by

SFnij ¼ �

3ffiffiffi2p

4rerd �

m

jmj

� �ij

jrd/ijjrd/ij (60)

where the discretization is similar to that in LSM.Figure 8 shows examples of the smoothed d function dað/Þ and

ejr/j2 for LSM and PFM, respectively. The level set / ¼ x andphase-field / ¼ tanhðx=ð

ffiffiffi2p

eÞÞ functions are given for eachmethod. The other parameters h ¼ 1=32, a ¼ 3h, ande ¼ 6h=½2

ffiffiffi2p

tanh�1ð0:9Þ� are used.

6.1.3 IBM. In IBM, the discretization of the force density fn

is defined by the boundary configuration Xn. For l ¼ 1;…;N,

fnl ¼

rDslþ1

2

Xnlþ1 � Xn

l

Dslþ1

� Xnl � Xn

l�1

Dsl

� �(61)

where Dsl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxl � xl�1Þ2 þ ðyl � yl�1Þ2

qis the Lagrangian mesh

size and r is a surface tension coefficient. Here, Dslþ12

¼ ðDsl þ Dslþ1Þ=2. For implementation, because the interface isclose, Xn

0 ¼ XnN and Xn

Nþ1 ¼ Xn1. The boundary force is then

spread into the nearby lattice points of the fluid:

SFnij ¼

XN

l¼1

fnl d

2hðxij � Xn

l ÞDslþ12

(62)

for i ¼ 1;…;Nx and j ¼ 1;…;Ny, where d2h is a two-dimensional

discrete d function:

d2hðxÞ ¼

1

h2w

x

h

w

y

h

(63)

where wðrÞ=h is a one-dimensional discrete d function. Commond functions include two-point, three-point, four-point, six-point,and four-point cosine functions [78,83,86]. The motivationfor this particular choice of the function wðrÞ is given inRefs. [68,78]. The formulas that have been applied in the literatureare as follows: two-point [93], three-point [76,87], four-point[90,91], six-point [73,84], and four-point cosine [68,71,80,89].wðrÞ for the d functions are as follows:

two-point function

wðrÞ ¼ 1� jrj; if jrj � 1;0; otherwise

�(64)

three-point function

wðrÞ ¼

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�3r2 þ 1p

3; if jrj � 1

2;

5� 3jrj �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�3ð1� jrjÞ2 þ 1

q6

; if1

2� jrj � 3

2;

0; otherwise

8>>>>>><>>>>>>:

(65)

four-point function

wðrÞ ¼

3� 2jrj þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4jrj � 4r2

p8

; if jrj � 1;

5� 2jrj �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�7þ 12jrj � 4r2

p8

; if 1 � jrj � 2;

0; otherwise

8>>>>><>>>>>:

(66)

Fig. 8 Smoothed d functions for LSM and PFM with h 5 1=32,a 5 3h, and e 5 6h=½2

ffiffiffi2p

tanh�1ð0:9Þ�

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six-point function

wðrÞ ¼

61

112� 11

42jrj � 11

56r2 þ 1

12jrj3 þ

ffiffiffi3p

336ð243þ 1584jrj � 748r2

�1560jrj3 þ 500r4 þ 336jrj5 � 112r6Þ1=2; if 0 � jrj � 1;

21

16þ 7

12jrj � 7

8r2 þ 1

6jrj3 � 3

2wðjrj � 1Þ; if 1 � jrj � 2;

9

8� 23

12jrj þ 3

4r2 � 1

12jrj3 þ 1

2wðjrj � 2Þ; if 2 � jrj � 3;

0; otherwise

8>>>>>>>>>>>><>>>>>>>>>>>>:

(67)

four-point cosine function

wðrÞ ¼1þ cosðpr=2Þ

4; if jrj � 2;

0; otherwise

8<: (68)

The discrete d functions may produce nonphysical force oscilla-tions when dealing with moving boundaries or free interfaces.However, the oscillations can be controlled by careful selection of

the discrete d function [82]. In Ref. [86], the authors carried outthe stability analysis of the feedback forcing scheme for thevirtual boundary method for various types of regularized d func-tions. They showed that the stability regime could be widen byincluding more supported points for the d function. A smoothingtechnique was developed to construct the discrete functions fromregular ones, which can significantly reduce the nonphysical oscil-lations [88]. For example, a smoothed four-point function w canbe written as

wðrÞ ¼

12þ p� 8r2

32; if 0 � jrj � 1

2;

2þ ð1� jrjÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2þ 8jrj � 4r2

p8

�arcsin

ffiffiffi2pðjrj � 1Þ

� �8

; if1

2� jrj � 3

2;

68� p� 48jrj þ 8r2

64þ ðjrj � 2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�14þ 16jrj � 4r2

p16

þarcsin

ffiffiffi2pðjrj � 2Þ

� �16

; if3

2� jrj � 5

2;

0; otherwise

8>>>>>>>>>><>>>>>>>>>>:

(69)

Figure 9(a) shows the regularized d functions associated withEqs. (64)–(67) and Fig. 9(b) depicts the four-point d functionsassociated with four-point, four-point cosine, and smoothed four-point functions.

6.2 Variable Density and Viscosity. In this section, we pres-ent the approximations for the variable density and viscosity.

6.2.1 LSM. In LSM, we define the variable density and vis-cosity by

qnij ¼ q2 þ q1 � q2ð ÞHað/n

ijÞ (70)

gnij ¼ g2 þ g1 � g2ð ÞHað/n

ijÞ (71)

where Hað/Þ is from Eq. (25).

6.2.2 PFM. In PFM, we define the variable density and vis-cosity by

qnij ¼ q2 þ q1 � q2ð Þ

1þ /nij

2(72)

gnij ¼ g2 þ g1 � g2ð Þ

1þ /nij

2(73)

6.2.3 IBM. We present an algorithm for the discrete indicatorfunction [74,79,90]. Let Gn

ij be the discretization of the right-handside of Eq. (32):

Gnij ¼

XN

i¼1

nnl d

2hðxij � Xn

l ÞDslþ12

(74)

where Dslþ12¼ Ds1 þ Dslþ1ð Þ=2. To calculate the unit normal vec-

tor nnl at Xn

l , we use a quadratic polynomial approximation with

three points Xnl�1, Xn

l , and Xnlþ1. First, let xðsÞ ¼ a1s2 þ b1sþ c1

and yðsÞ ¼ a2s2 þ b2sþ c2, and assume ðxð�DslÞ; yð�DslÞÞ¼ Xn

l�1, ðxð0Þ; yð0ÞÞ ¼ Xnl , and xðDslþ1Þ; yðDslþ1Þð Þ ¼ Xn

lþ1. We

then calculate the coefficients a1, b1, and c1 as follows:Fig. 9 (a) Regularized delta functions and (b) Four-point dfunctions.

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a1

b1

c1

0B@

1CA ¼

Ds2l �Dsl 1

0 0 1

Ds2lþ1 Dslþ1 1

0B@

1CA�1

xðDslÞxð0Þ

xðDslþ1Þ

0B@

1CA (75)

And the coefficients a2, b2, and c2 are similarly calculated.Finally, the unit normal vector is approximated by

nl ¼ðys;�xsÞðxs; ysÞj j

s¼0

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

1 þ b22

q ðb2;�b1Þ (76)

We then solve Poisson’s equation using the multigrid method[104] with homogeneous Dirichlet boundary conditions:

DdHnij ¼ rd � Gn

ij (77)

In IBM, we can define the variable density and viscosity by

qnij ¼ q2 þ q1 � q2ð ÞHn

ij (78)

gnij ¼ g2 þ g1 � g2ð ÞHn

ij (79)

6.3 Update the Interface. In this section, we describe thediscretization of the interface advection process.

6.3.1 LSM. The evolution of level set / is given by

/nþ1 ¼ /n � Dtðunþ1 � r/nÞ (80)

The convective term is discretized as

unþ1 � r/n� �

ij¼

unþ1iþ1

2;jþ unþ1

i�12;j

2h�/n

iþ12;j� �/n

i�12;j

þvnþ1

i;jþ12

þ vnþ1i;j�1

2

2h�/n

i;jþ12� �/n

i;j�12

(81)

where the edge values �/ni61

2;j and �/n

i;j612

are computed using the

essentially nonoscillatory (ENO) scheme derived in Ref. [7]. For

example, the procedure for computing the quantity �/niþ1

2;jis as

follows:

k ¼i; if unþ1

iþ12;j 0;

iþ 1; otherwise

((82)

a ¼/n

kj � /nk�1;j

h(83)

b ¼/n

kþ1;j � /nkj

h(84)

d ¼a; if jaj � jbj;b; otherwise

�(85)

�/niþ1

2;j ¼ /n

ij þh

2dð1� 2ðk � iÞÞ (86)

The quantities �/ni;j�1

2and �/n

i;j612

are computed in a similar manner.

6.3.2 PFM. The phase-field function /n is evolved by apply-ing the nonlinear splitting stabilized (NLSS) scheme [105,106] tothe Cahn–Hilliard equation:

/nþ1 � /n

Dt¼ �rd � ð/nunþ1Þ þMDdl

nþ1 �MDd/n (87)

lnþ1 ¼ ð/nþ1Þ3 � e2Dd/nþ1 (88)

where

rd � ð/nunþ1Þij ¼1

2h/n

iþ1;j þ/nij

unþ1

iþ12;j� /n

ij þ/ni�1;j

unþ1

i�12;j

þ 1

2h/n

i;jþ1 þ/nij

vnþ1

i;jþ12

� /nij þ/n

i;j�1

vnþ1

i;j�12

(89)

and

Dd/nij ¼

/niþ1;j þ /n

i;jþ1 � 4/nij þ /n

i�1;j þ /ni;j�1

h2(90)

We use a nonlinear full approximation storage (FAS) multigridmethod [49]. The nonlinearity is treated using one step of New-ton’s iteration. A pointwise Gauss–Seidel relaxation scheme isused as the smoother in the multigrid method [104].

6.3.3 IBM. Once the updated fluid velocity unþ1 has beendetermined, we can find the velocity Unþ1 and then the new posi-tion Xnþ1 of the immersed boundary points. This is done using adiscretization of Eqs. (9) and (10). That is, for l ¼ 1;…;N,

Unþ1l ¼

XNx

i¼1

XNy

j¼1

unþ1ij d2

hðxij � Xnl Þh2 (91)

Xnþ1l ¼ Xn

l þ DtUnþ1l (92)

7 Numerical Examples

In this section, we perform the following numerical examples,calculating the pressure difference, comparing the droplet defor-mation under shear flow, and simulating the falling droplet usingeach method. Throughout the experiments, we use the surface ten-sion formula [17] for PFM and four-point function [66] for IBM.We take five reinitialization steps for LSM. The results show thatfive iterations are enough to accurately recover the distancefunction.

7.1 Pressure Difference. We theoretically and numericallycalculate the pressure difference of a droplet. In the absence ofviscous, gravitational, and other external forces, the pressuregradient is balanced by the surface tension force rp ¼ SF. ByLaplace’s formula for an infinite cylinder surrounded by a back-ground fluid at zero pressure, the exact pressure difference is½p�e ¼ r=r, where r is the droplet radius and r is a surface tensioncoefficient [5]. The numerical pressure difference ½p�h is calcu-lated by

½p�h ¼ maxxij2Xh

pij � minxij2Xh

pij (93)

where pij is the pressure in the computational domain Xh.For the numerical simulation, the droplet is placed at the center

of the unit domain X ¼ ð0; 1Þ � ð0; 1Þ and its radius is r ¼ 0:25.The pressure differences are computed with h ¼ 1=2n, for n ¼ 5,6, and 7. We employ r ¼ 5 with the constant density and viscosityq ¼ 1 and g ¼ 1. Therefore, the pressure difference is ½p�e ¼ 20.For example, using the 64� 64 mesh grid, numerical results forthe pressure are shown in Fig. 10. For other parameters, we seta ¼ 0:03 in LSM, e ¼ 0:03 in PFM, and N ¼ 800 in IBM.

Table 1 lists the numerical pressure difference ½p�h with a dif-ferent mesh grid for each method. The numerical pressure differ-ence ½p�h reaches a qualitative agreement with the theoreticalvalue as the mesh is refined.

7.2 Deformation of Droplet Under Shear Flow. We simu-late the droplet deformation under shear flow. A single droplet X1

in an ambient fluid X2 between two parallel plates is

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schematically depicted in Fig. 11. The shear flow is generated bythe Dirichlet boundary condition u ¼ ð6 _cy; 0Þ at the top and bot-tom boundary, where _c is the shear rate. We use the periodic con-dition for the x-direction.

We consider the density and viscosity ratios of the droplet toambient fluid to have unit values and neglect the influence of thegravitational force. This problem is nondimensionalized by theundeformed droplet radius R. We define the Reynolds numberRe ¼ q2R2 _c=g2 and capillary number Ca ¼ g2R _c=r. The Webernumber can then be written as We ¼ Re � Ca. The Navier–Stokesequations (1) and (2) are nondimensionalized as

@u

@tþ u � ru ¼ �rpþ 1

ReDuþ 1

WeSF (94)

r � u ¼ 0 (95)

To measure the magnitude of the droplet deformation, we usethe Taylor deformation number D, defined as D ¼ ðL� BÞ=ðLþ BÞ, where L and B are the major and minor droplet semiaxes,shown in Fig. 11.

First, we compare the deformation number of the three methodswith the results of Ref. [72]. For the initial state, the droplet radiusis R ¼ 0:5, centered at ð0; 0Þ in the computational domainX ¼ ð�1; 1Þ � ð�1; 1Þ. The shear rate _c ¼ 1, 32� 32 mesh grid,and time step Dt ¼ 0:0005 are used. For other parameters, a ¼ his used in LSM, e ¼ 0:07 and Pe ¼ 1=e are used in PFM, andN ¼ 402 is used in IBM. Figure 12 shows the deformation number

with different Re and Ca numbers. The results are in a good agree-ment with those studied in Ref. [72].

Next, we compare the results of the three methods by the longevolution. For the initial state, the droplet radius is R ¼ 0:5 and itscenter is ð0; 0Þ in X ¼ ð�2; 2Þ � ð�1; 1Þ. We take the otherparameters as: Re ¼ 5, Ca ¼ 0:6, and _c ¼ 1. The 256� 128 meshgrid and time step Dt ¼ 0:0002 are used. For LSM, the parametera ¼ 3h is used. For PFM, a Peclet number Pe ¼ 0:1=e ande ¼ 0:06 are used. For IBM, we fix the number of Lagrangianmarker points as 1000. Figure 13(a) shows that the droplet shapesat the time t ¼ 8. These shapes are in qualitative agreement foreach result. We also plot the deformation number D with the timet in Fig. 13(b). The difference between the curves for all methodsis small at all times.

7.3 Simulation of the Falling Droplet. Simulating a fallingdroplet is a classical test problem in two-phase flows. We demon-strate the adaptability of the three methods by simulating a fallingdroplet X1 surrounded by the ambient fluid X2 under a gravita-tional force. We use the periodic and zero Dirichlet boundary con-ditions for the x- and y-directions of velocity field, respectively.

We nondimensionalize the governing equation based on theproperties of the ambient fluid, including the dimensionless pa-rameters, Reynolds number Re ¼ q2UcLc=g2, Weber numberWe ¼ q2U2

c Lc=r, and Froude number Fr ¼ U2c=ðgLcÞ, where Uc is

the characteristic velocity, Lc is the characteristic length, and q2,g2 are the density and viscosity of the ambient fluid, respectively.We consider the viscosity ratio of the droplet and ambient fluid tohave the unit value. The nondimensionalized Navier–Stokes equa-tions (1) and (2) can be written as

q@u

@tþ u � ru

� �¼ �rpþ 1

ReDuþ 1

WeSFþ q

Frg (96)

r � u ¼ 0 (97)

Fig. 10 Numerical pressure field p for (a) LSM, (b) PFM, and (c) IBM

Fig. 11 Schematic illustration of droplet deformation undershear flow

Fig. 12 Deformation number D as a function of time t compar-ing with results (solid lines) in Ref. [72]

Table 1 Pressure difference ½p�h with r 5 5 and r 5 0:25 fordifferent mesh sizes

Mesh 32� 32 64� 64 128� 128

LSM 20.491 20.118 20.057PFM 17.681 19.441 19.984IBM 20.066 20.035 20.020

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We set the droplet density to q1 ¼ 3 and ambient fluid density toq2 ¼ 1. We consider a circular droplet with radius R ¼ 0:25centered at ð0:5; 1:5Þ in the computational domain X¼ ð0; 1Þ � ð0; 2Þ. We take the characteristic parameters asUc ¼ Lc ¼ 1, and set r ¼ 0:04 and g1 ¼ g2 ¼ 0:01, then thedimensionless parameter values are Re ¼ 100, We ¼ 25, andFr ¼ 1=g. The gravitational force is set to g ¼ ð0;�1Þ. The128� 256 mesh grid, space step h ¼ 1=64, and time stepDt ¼ 0:0002 are used. We consider the effect of the gravitationalforce together with the relatively small interfacial tension.

In LSM, the parameter a in the smoothed Heaviside functiondetermines the interfacial thickness for the transition of fluid prop-erties. An appropriate value of a will give a reasonable droplet

evolution. Figure 14 shows the evolution of the droplet shape fordifferent values of a. For a narrow interface a ¼ 0:5h, the dropletshrinks compared to those of the other parameters a ¼ h and 2h. Itcan also be seen that there is no evident shape difference for largervalues of a ¼ h, 2h. Our comparison can suggest that a ¼ 1:5h isaccurate enough.

In PFM, the Peclet number plays an important role in the inter-face evolution. To investigate the effect of the Peclet numberand determine an appropriate value, we consider three differentPeclet numbers Pe ¼ 0:1=e, 1=e, and 10=e, where e ¼ 4h=½2

ffiffiffi2p

tan�1ð0:9Þ�. The droplet contours at different times and Pec-let numbers are shown in Fig. 15. The droplet shape is flatter inthe case of Pe ¼ 0:1=e, while the diffuse interface is smeared outor compressed in the case of Pe ¼ 10=e. These could lead to inac-curate and nonphysical results. Compared with the results ofLSM, Pe ¼ 1=e is a better choice.

In IBM, we consider the effect of the distribution of markerpoints on the evolution of droplet. The initial marker points areevenly distributed as follows: Let N be the nearest integer numberof 2pRm=h, where m is an integer. Dsl is approximately taken as

Fig. 15 Evolution of the droplet using different Peclet number(a) Pe 5 0:1=e, (b) Pe 5 1=e, and (c) Pe 5 10=e in PFM. The con-tours are drawn at times t 5 0, 2.5, and 4.5 from top to bottom.Contour levels are �0:9, �0:45, 0, 0.45, and 0.9.

Fig. 16 Evolution of the droplet in IBM (a) without and (b) withdeletion and addition procedures. The marker points are drawnat times t 5 0, 2:5, and 4:5 from top to bottom.

Fig. 14 Evolution of the droplet using different interfacialthickness parameter (a) a 5 0:5h, (b) a 5 h, and (c) a 5 2h inLSM. The contours are drawn at times t 5 0, 2.5, and 4.5 fromtop to bottom. Contour levels are �2h, �h, 0, h, and 2h.

Fig. 13 (a) Interfaces of the three methods and (b) Deformationnumbers as a function of time.

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h=m. We use a moderate number m ¼ 4 and, therefore, N ¼ 804for the initial droplet. Figure 16(a) shows the time evolution ofdroplet shape with a fixed number N. Note that Dsl is varied bythe flow field. The droplet shape is well maintained up to t ¼ 2:5,but marker points become concentrated at t ¼ 4:5, making theconnected line heavily intertwined and eventually leading to acollapse of the droplet shape. To overcome this problem, we adap-tively add and delete marker points so that 0:1h < Dsl < 0:5h.Figure 16(b) shows the evolution of droplet shape with the dele-tion and addition procedures. The number of marker points as afunction of time is shown in Fig. 17. The interface points are moreevenly distributed and the droplet shape remains good at t ¼ 4:5in Fig. 16(b).

Finally, we compare the droplet evolution using the optimalstrategy of each method. Figure 18 shows a comparison of thedroplet shape at time t ¼ 0, 2:5, and 4:5 for three methods. Asshown in the figure, the droplet shapes are well matched at each

time, demonstrating their strong ability for accurate interfacialrepresentation.

7.4 Computational Cost. We compare the computationalcost of the shear flow and falling droplet simulations for eachmethod. The initial conditions and parameters follow those ofFigs. 13 and 18. Table 2 lists the central processing unit (CPU)time of each method for calculating up to time t ¼ 3. We can seethat IBM has less computational cost than the other two methodsin the shear flow simulation. In this case, LSM and PFM haveadditional computational costs such as the reinitialization andCahn–Hilliard equations. For the falling droplet test, the necessityof solving an indicator function and redistributing marker pointsby adding and deleting points accounts for the rapid CPU time in-crement for IBM. Therefore, the CPU time cost of each method iscase by case, and the three methods have more or less similarcomputational expense.

8 Discussion and Conclusions

From the numerical point of view, the three methods we havedescribed belong to the “diffuse interface” model because of theuse of the d function. The smeared-out interface by the d functionis in favor of computation of surface tension and fluid propertieseven if it is a real sharp interface in the physical problems. For thepresentation of interface, there is a finite width for the phase-fieldmethod, but it is “zero” width for the level set and immersedboundary methods. Meanwhile, the interface is implicitlyexpressed by the level set and phase-field methods and explicitlyby the immersed boundary method.

Because the interface is represented by an implicit level setfunction, there is no need for special consideration of complexinterfacial topology changes with separation and merging. For thelevel set method using reinitialization technique, the accuracy andefficiency are greatly determined by that of reinitialization proce-dure, and mass loss is a problem for the original level set method.For the phase-field method, it shares the advantages with the levelset method that there is no need for tracking the interface explic-itly. In the Cahn–Hilliard type phase-field model, we need to care-fully select the Peclet number to make interface evolutionreasonable, and an appropriate interface width is also needed. Forthe immersed boundary method, interface can be representedaccurately and explicitly by using a great number of Lagrangianmarker points. However, the distribution of marker points must becarefully considered in case nonphysical or undesirable phenom-ena occur.

In this paper, we applied the level set, phase-field, andimmersed boundary methods to the computation of two-phaseflows governed by the incompressible and immiscible Navier–Stokes equations. We explained the numerical discretization ofeach method, introduced their essential concepts, and discussedthe strengths and weaknesses. We then performed a series of nu-merical experiments using each method. Using a pressure differ-ence experiment, we validated each of the algorithms, showingthat the numerical solutions are qualitatively in agreement withthe theoretical value. By simulating the droplet deformation undera shear flow, the three methods were preliminarily verified.Finally, in the challenging simulation of a falling droplet, we dis-cussed in detail the effects of different parameter values on theevolution of the droplet. CPU time cost was also considered intwo simulations to compare the efficiency of each method. The

Fig. 18 Comparison of the falling droplet for the three meth-ods with optimal parameters at time t 5 0, 2:5, and 4:5

Table 2 CPU times (s) for the shear flow and falling dropletsimulations. The calculations are run to the time t 5 3.

Case LSM PFM IBM

Shear flow 2018 2849 421Falling droplet 2320 4538 3234

Fig. 17 Variation of the number of marker points as a functionof time in case of Fig. 16(b)

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results agreed reasonably well, demonstrating the ability of allthree methods to capture the interface of two-phase flows.

Acknowledgment

The corresponding author (J.S. Kim) was supported by theMinistry of Science, ICT & Future Planning (MSIP) (NRF-2011-0023794). The authors are grateful to the anonymous refereeswhose valuable suggestions and comments significantly improvedthe quality of this paper.

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021301-14 / Vol. 136, FEBRUARY 2014 Transactions of the ASME

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